Slide 1

Challenge Paper: Marginal Probabilities for Instances and Classes

Oliver SchulteSchool of Computing ScienceSimon Fraser UniversityVancouver, Canada

#/12This paper presents a challenge, not a solution.1Class-Level and Instance-Level QueriesClassic AI research distinguished two types of probabilistic relational queries. (Halpern 1990, Bacchus 1990). Halpern, An analysis of first-order logics of probability, AI Journal 1990.Bacchus, Representing and reasoning with probabilistic knowledge, MIT Press 1990.Relational QueryClass-level QueryReference ClassWhat is the percentage of flying birds?BirdsWhat is the percentage of friendship pairs where both are women?Pairs of FriendsWhat is the percentage of A grades awarded to highly intelligence students?Student-course pairs where student is registered in course.Instance-Level QueryGiven that Tweety is a bird, what is the probability that Tweety flies?Given that Sam and Hilary are friends, and given the genders of their other friends, what is the probability that Sam and Hilary are both women?What is the probabiity that Jack is highly intelligent given his grades?Instance-level queriesGround factsType 2 probabilitiesClass-level queriesRelational StatisticsType 1 probabilities#/12M. Chiang and D. Poole (2012), Reference classes and relational learning2A connection between class-level and instance-level probabilitiesMarginal Probabilities for Instances and Classes

Percentage of Flying Birds = 90%. Halpern: Probability that a typical or random bird flies is 90%.

What is the answer to P(Flies(Tweety))? It should be 90%!#/12If the only thing you know about Tweety is that Tweety is a bird, you should use the class frequencies.Using logical predicate notation.3Halperns Instantiation VersionGiven that Tweety is a bird (and nothing else), the probability that Tweety flies =the probability that a randomly chosen bird flies.

P(Flies(Tweety)|Bird(Tweety)) = P(Flies(B)|Bird(B)).Assuming that 1st-order variables and constants are typed:

P(Flies(Tweety)) =P(Flies(B)).The Marginal Equivalence Principle.#/12Instantiate B with Tweety4Marginal Probabilities for Instances and ClassesaFour Arguments for Marginal Equivalence#/12I: Intuitive PlausibilityUsed in cold-start problems.Equivalent to Millers principle.Marginal Probabilities for Instances and Classesa#/12II: Score MaximizationMarginal Probabilities for Instances and ClassesaCourseIDdifficultyP1(diff)P2(diff)100lo1/31/2200hi2/31/2300hi2/31/2400lo1/31/2Score4/270.15 1/160.17#/12III: Latent Variable Models Satisfy Marginal EquivalenceMarginal Probabilities for Instances and Classesaintelligence(S)diff(C)Registered(S,C)U(S)U(C)CourseIDdifficultyU(C)100lo10200hi20300hi15400lo12SNameint.U(S)Annalo30Bobhi20#/12IV: Something ElseMarginal Probabilities for Instances and Classesa#/12The ChallengeIf we accept that an SRL system should satisfy class-instance marginal equivalence, how do we design a system to achieve that?Marginal Probabilities for Instances and Classesa#/12Parametrized Bayes Net ExamplesMarginal Probabilities for Instances and Classesaintelligence(S)diff(C)Registered(S,C)Proposition If each node in the ground network has a unique set of parents, then class-level marginals = instance-level marginal.For other structures, it depends on the combining rule/parameters used.#/12What about aggregate functions, MLNs, dependency networks?11Constraints are GoodPedro Domingos: The search space for SRL algorithms is very large even by AI standards.Class-instance marginal equivalence reduces the search space.Strong theoretical foundation.The challenge is to implement the constraint.Marginal Probabilities for Instances and ClassesaParametrized Bayes Nets

PBN + Marginal Equivalence#/12